Econ 3120 1st Edition Lecture 3 Outline of Last Lecture I Marginal Distributions II Expectations and Variance III Covariance Outline of Current Lecture IV Covariance V Conditional Expectation VI Normal Distribution Current Lecture 2 3 Covariance When we are dealing with more than one random variable it is useful to summarize how these two random variables move together Suppose we have two random variables X and Y and we define E X x and E Y y The covariance between these two random variables is defined as Cov X Y xy E X x Y y We can also write the covariance as Cov X Y E X x Y E X Y y E XY x y 2 3 1 Properties of the covariance 1 If X and Y are independent then Cov X Y 0 This follows since E XY E X E Y when X and Y are independent 2 For any constants a1 b1 a2 b2 Cov a1X b1 a2Y b2 a1a2Cov X Y Now that we know about the Covariance we can define a third property of the variance 3 Var aX bY a 2Var X b 2Var Y 2abCov X Y One issue with covariance is that the units are difficult to interpret It turns out that we can scale covariance by the standard deviations of both variables and to get the unit friendly correlation Corr X Y xy Cov X Y sd X sd Y xy x y 2 4 Conditional Expectation In econometrics we often want to know how much one variable X tells us about another variable Y One way to do this is by using covariance and correlation but another concept we will be using a lot is conditional expectation Conditional expectation written as E Y X x often shortened to either E Y X or E Y x tells us the mean of Y conditional on some value of X The conditional expectation is defined in a similar way to the unconditional expectation above but using the conditional probability distribution functions discrete r v E Y x y y f y x continuous r v E Y x y y f y x Example Suppose we are studying the relationship between schooling and earnings and that hourly wages and schooling are our random variables How does the mean wage vary with the schooling level Our CEF conditional expectation function might look something like this E WAGE EDUC 4 0 6 EDUC These notes represent a detailed interpretation of the professor s lecture GradeBuddy is best used as a supplement to your own notes not as a substitute Thus for each level of schooling we know the mean wage Properties of the conditional expectation 1 If X and Y are independent then E Y X E Y 2 E E Y X E Y This is called the law of iterated expectations Example Suppose we want to know E WAGE in the example above and we know that E EDUC 11 5 Then by the law of iterated expectations E WAGE E E WAGE EDUC E 4 0 6 EDUC 4 0 6 E EDUC 4 0 6 11 5 10 9 3 Normal Distribution Because of its properties the normal distribution is probably the most commonly used distribution in statistics and econometrics We won t go through all of the properties of the normal here but they will become apparent as we go along The normal is described by the pdf f x 1 2 exp x 2 2 2 x The parameters are E X and 2 Var X We also write a normally distributed random variable X as X Normal 2 One special case of the normal is the standard normal which has mean 0 and variance 1 z f z 1 2 exp z 2 2 z We often convert normal random variables to standard normal because the values of the standard normal are more easily computed The standard normal cumulative distribution function is denoted by z If we can convert a variable to standard normal and express probabilities in terms of z we can look up the values in a table Some useful properties of the normal distribution 1 The standard normal is symmetric about the origin That is for a positive constant c c c or c 1 c 2 If X Normal 2 then X Normal 0 1 3 If X Normal 2 then aX b Normal a b a 2 2 4 Any linear combination of independent identically distributed i i d normal random variables has a normal distribution Example 1 if X1 X2 X3 are i i d random variables distributed as Normal 2 what is the distribution of X1 2X2 X3